First we must put \( 1 + i \) in trigonometric form. One may use either degree or radian measure, but you have already seen an example using radian measure so in this solution we will switch to degree measure.
The modulus is \(\sqrt{2}\) and, since the complex number \( 1 + i \) lies on the terminal side of the angle \(45^\circ\) then its angle (also called its argument) is \(45^\circ\).
The seventh power of the complex number will have a modulus which is the seventh power of \(\sqrt{2}\), that is \(\left(\sqrt{2}\right)^7=8\sqrt{2}\).
The angle of the seventh power will be seven times the angle \(45^\circ\) which is \(315^\circ\). Thus, the seventh power of \(( 1 + i )\) is
\( (1+i)^7=8\sqrt{2} ( \cos 315^\circ + i \sin 315^\circ ) = 8\sqrt{2}\left(\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2} i\right)=8 – 8 i\)